A novel method is presented for efficiently testing the stability of an object under gravity and contact forces, that accommodates empirical determination of the set of admissible forces exerted at contacting pairs of surfaces. These admissible force volumes may exhibit a wide variety of geometries, including anisotropy, adhesion, and even non-convexity. The method discretizes the contact region into patches, performs a convex decomposition of a polyhedral approximation to each admissible force volume, and then formulates the problem as a mixed integer linear program. The model can also accommodate articulated robot hands with joint torques, joint frictions, and spring preloads. Predictions of our method are evaluated experimentally in object lifting tasks using a gripper that exploits microspines to exert strongly anisotropic forces.